Travelling Waves For A Generalized Fisher Equation With The Higher Order Terms As Perturbations

This paper is concerned with the traveling wavefronts and spreading speed of a generalized Fisher equation which contains both sixth and fourth order spatial derivative terms as perturbations. By the geometric singular perturbation theory combining with linear chains technology and Fredholm theory, we first establish the existence of such wavefronts when the sixth and fourth order terms have sufficiently small coefficients. And the effects of the higher derivative terms are discussed when the minimal wave speed is involved, which implies that sixth order spatial derivative term can increase the wave speed while the forth one can decrease the minimal wave speed. In addition, with the aid in a kind of integral transformation, we give the precise approaching behavior of the wavefronts, and this result will enrich the understanding about the travelling waves broadly . Then, we prove that the wavefronts are locally stable in a proper weighted functional space, which is based on the construction of different energy functionals. The stability of traveling wavefronts indicates that the traveling wavefronts are useful in understanding the corresponding initial value problem.